In Apostol's Mathematical Analysis (second edition), it is written, on p.3:
The real numbers are often represented geometrically as points on a line (called the real line or the real axis). A point is selected to represent $0$ and another to represent $1$ [...]. This choice determines the scale. Under an appropriate set of axioms for Euclidean geometry, each point on the real line corresponds to one and only one real number and, conversely, each real number is represented by one and only one point on the line. It is customary to refer to the point $x$ rather than the point representing the real number $x$.
Question: Is there a formal proof of the bold sentence? If so, where can I find such a proof?